Four-photon correction in two-photon Bell experiments

0.25 MB·English

by  Valerio Scarani

Four-photon correction in two-photon Bell experiments
Valerio Scarani, Hugues de Riedmatten, Ivan Marcikic, Hugo Zbinden, Nicolas Gisin
Group of Applied Physics, University of Geneva
20, rue de l’Ecole-de-M´edecine, CH-1211 Geneva 4, Switzerland
February 9, 2008
Abstract
5
0
Correlated photons produced by spontaneous parametric down-conversion are an essential tool for
0
2 quantumcommunication,especiallysuitedforlong-distanceconnections. Tohaveareasonablecountrate

n afterallthelossesinthepropagation andthefiltersneededtoimprovethecoherence,itisconvenientto
a
J increasetheintensityofthelaserthatpumpsthenon-linearcrystal. Bydoingso,however,theimportance

6
of the four-photon component of the down-converted field increases, thus degrading the quality of two-
2
photon interferences. In this paper, we present an easy derivation of this nuisance valid for any form

2
v of entanglement generated by down-conversion, followed by a full study of the problem for time-bin
9 entanglement. We find that the visibility of two-photon interferences decreases as V = 1−2ρ, where
8
1 ρ is, in usual situations, the probability per pulse of creating a detectable photon pair. In particular,
7
the decrease of V is independent of the coherence of the four-photon term. Thanks to the fact that ρ
0
4 can bemeasured independentlyof V, theexperimental verification of our prediction is providedfor two
0
/ different configuration of filters.
h
p
-
t
n 1 Introduction
a
u
q The distribution of a pair of entangled photons to two distant partners is the building block of quantum
:
v
communicationprotocols[1,2]. Theentangledphotonsareproducedbyparametricdown-conversion(PDC)
i
X
in a non-linear crystal. As well-known,this process creates pairs of photons at the first order; but when the
r
a
pumping intensity increases, four- and more-photon components become important in the down-converted
field [3, 4]. If one can post-select the number of photons, higher-photon components of the field may
turn out to be a useful resource (an ”entanglement laser”, see [5]). In other cases however, these higher-
number components turn out to be quite a nuisance. In particular, one is often interested in two-photon
phenomena, just think of the Bell-state measurement (BSM) that is needed in teleportation. The presence
of higher-number components obviously degrades the quality of the two-photon interferences. In long-
distance implementations, one can hardly overcome this nuisance by working with low pump intensities:
afterpropagationalongseveralkilometersoffibers,manyphotonsarelostbecauseofthelossesinthefibers,
andtheefficiencyofthedetectorsattelecomwavelengthsislow,typically10%. Moreover,ifthetwophotons
1
comefromdifferentsourcesandhavetointerfereatabeam-splitter(asisthecasefortheBSM),filtersmust
be introduced to ensure coherence. Thus, in order to have a reasonable count rate, one has to increase
the pump power — and this unavoidably increases the number of unwanted higher-number components. In
this paper, we address the degradation of the visibility of two-photon interferences due to the presence of
four-photon components in the field, thus completing the partial study provided in Ref. [6].
InSection2,wegiveaneasyderivationofthetopicandtheresultsthatisvalidforanyformofentanglement
generated by down-conversion, under the assumption that the four-photon component is described by two
independentpairs. Intherestofthepaper,werelaxthatassumption: indeed,thefour-photoncoherencecan
vary from zero (two independent pairs) to one (state of single-mode down-conversion [7]) according to the
experimentalconditions[8,9,10]. Weprovethatthelossofvisibilitydoesnotdependonthecoherenceofthe
four-photon state, but only on a parameter ρ that is basically the probability of creating a detectable pair.
Forthisfullstudy,weshallfocusontime-bin entanglement,aformofentanglementthatismorerobustthan
polarization for long-distance applications in optical fibers. Visibilities large enough to allow the violation
of Bell’s inequalities for two photon [6, 11], quantum cryptography[12], and long-distanceteleportation[13]
have been demonstrated in the recent years for this form of entanglement. For time-bin entanglement, the
present study requires the multimode formalism, introduced in Section 3. In fact, as the name suggests, a
time-bin qubit is a coherent superposition of two orthogonal possibilities, the photon being at a given time
t = 0 (first time-bin) or at a later time t = τ (second time-bin). Separate time-bins must be created by a
pump field consisting of separate pulses: the finite temporal size (thence, the non-monochromaticity)of the
pump pulses and the down-convertedphotons is a necessary feature of time-bin qubits.
In Section 4, we describe a setup that is used for measure the parameter ρ. In Section 5, we introduce
the setup for measuring time-bin entanglement(a Fransoninterferometer with a suitable source)and derive
our main prediction, namely the decrease of visibility due to the presence of four-photon components in the
state. In Section 6 we describe the experimental verification of our predictions. Section 7 is a conclusion.
Forreadability,thetechnicalitiesoftheformalismusedinSections4and5areleftforanAppendix. Wenote
that the calculations of the two-photon coincidence rate provides the first explicit calculation of time-bin
Bell experiments using the full formalism of quantum optics.
2 Easy derivation for incoherent four-photon component
The purpose of this Section is to derive the main results from a simple formalism, in order to gain intuition
aboutthephysicsoftheproblem. ThecontentofthisSectiondoesnotapplyonlytotime-binentanglement,
but to any formofentanglementobtainedby down-conversion,be it witha cw orwith a pulsed pump laser.
The probabilities that we are going to introduce in this Section are ”per detection window”. In the case of
a cw pump, this means ”per time resolutionof the detector”; in the case of a pulsed pump, this means ”per
pump pulse” (”per qubit”, in the language of time-bin entanglement [14]).
The calculation is possible in simple terms if we neglect the coherence of the four-photonterm, and assume
thatwhen four photonsare produced,they formtwoindependent pairs. The processofcreationofindepen-
2
a b
(I) 2 α a b β 2
PDC
a b
1 F F 1
A B
∆ ∆
a b
(II)
ω ω
∆ a ∆ b
A B
Figure 1: (I) Experimental setup to measure two-photon interferences, and meaningful parameters. Grey
letters: spatial modes in the fibers; PDC: parametric down-conversions; F: filters. (II) Spectral widths of
the down-converted photons (curves) and filters (grey shadows). Both filters are centered in the spectrum
of the down-converted photons, and their width are ∆ ∆ .
A B
≤
dent items obeys the Poissonian statistics: if P is the probability of creating a pair, we have P 1P2.
2c 4c ≃ 2 2c
For the setup, we refer to Fig. 1. We define ∆ as the spectral width of the photons in mode a, resp.
a,b
b, after down-conversion, that is, before the filters; the spectral width of the pump is denoted ∆ . As for
p
the filters, we suppose that they are centered in the spectrum of the down-converted photons, and that
they satisfy ∆ ∆ to avoid trivialities; furthermore, we suppose ∆ >>∆ , so that twin photons
A,B a,b A,B p
≤
certainlypassbothfilters,and∆ ∆ . Let’sfollowthe two-andthe four-photoncomponentthroughthe
B A
≥
setup, until the coincidence detection in modes a and b .
1 1
Two-photon component. To have a detection, both photons must pass the filters; because of the correlation
in energy, if photon a passes through F (that happens with probability ∆ /∆ ), then certainly photon
A A a
∼
b will pass through F , because this filter is larger and the photons are correlated in energy. The photons
B
are twins, therefore they interfere. Consequently, the detection rate due to two-photon components is (up
to multiplicative factors)
∆ 1
A
R = P 1+cos(α+β) . (1)
2 2c
∆ 2
a (cid:2) (cid:3)
Four-photon component. Once four photons have been produced, four two-photon coincidence events are
possible: two events in which we detect photons belonging to the same pair, and two events in which we
detectphotonsbelongingtodifferentpairs. Thefirstcaseissimilartothecaseoftwo-photons. Inthesecond
case, however, the fact that photon a passes its filter does not guarantee at all that photon b will do it as
well; and of course, no interference will take place. All in all,
∆ 1 ∆ ∆ 1
A A B
R = P 2 1+cos(α+β) + 2 . (2)
4 4c
(cid:26) ∆ 2 ∆ ∆ 2(cid:27)
a (cid:2) (cid:3) a b
ThetotalcountrateisthereforeR +R =R¯ 1 1+V cos(α+β) whereR¯ P ∆A andwherethe visibility
2 4 2 ≃ 2c∆a
(cid:2) (cid:3)
V is
1 ∆
V = = 1 P B + O(P2). (3)
1+ 1+PP2c2c ∆∆Bb − 2c ∆b 2c
3
Recallthat∆ issingledoutbytherelation∆ ∆ . Asexpected,V decreasesifP (proportionaltothe
B B A 2c
≥
pump power) is increased. Note also that ∆B 1: for a given pump power, the visibility increases if filters
∆b ≤
are in place. This is intuitive, considering the emission of two pairs: conditioned to the fact that photon in
mode a has passed the filter F , a photon passing F is more likely to be its twin (whose frequency must
A B
lie within the filter) than an uncorrelated photon (whose frequency may lie everywhere in the spectrum).
Finally, if only one filter is in place, then ∆B =1 and we recover the discussion presented in Ref. [6].
∆b
However,we are not really interested in fixing the pump power: rather, we’d like to fix the coincidence rate
at the detection R¯. Obviously, this means that if we narrow the filters, we must increase the pump power
in order to keep the coincidence rate constant. Strictly speaking, the quantity P ∆B is the probability per
2c∆b
qubit of creating a photon pair such that the photon in mode b passes through the (larger) filter. However,
∆A ∆B holds in magnitude for typicaldown-conversionprocesses andfilters; consequently,P ∆B R¯ is
∆a ≃ ∆b 2c∆b ≃
an estimate of the probability of creating a detectable pair.
The results of this Section are based on the assumption that the four-photon state is always described
by two independent pairs. Note that this assumption is certainly good in the case of cw pump, because
the time resolution of the detector is much larger than the coherence time of the down-converted photons.
The assumption is more questionable in the case of a pulsed pump. The rest of the paper shows, focusing
specifically on time-bin entanglement, that the degradation of visibility (3) is actually independent of the
coherence of the four-photon term.
3 General approach
3.1 The state out of down-conversion
The formalism to describe multimode down-conversion was introduced in Refs [15, 16] for the two-photon
component,andextendedtothefour-photoncomponentfortype-Idown-conversionin[17]. Wehaveapplied
this formalism to our case in Ref. [9]; we summarize here the main notations and results.
Thepumpfieldisassumedto beclassical,composedoftwoidenticalbutdelayedpulses: P(t)= I p(t)+
p
p (cid:0)
p(t+τ) , so in Fourier space
(cid:1)
P˜(ω) = I p˜(ω) 1+eiωτ . (4)
p
p (cid:0) (cid:1)
We use colinear type-I down-conversion in a non-degenerate regime ω = ω ; therefore, the signal and the
s i
6
idler photons can be coupled into different spatial modes a and b using a wavelength division multiplexer
(WDM). The phase-matching function is written Φ(ω ,ω ); we don’t need its explicit form in what follows.
a b
For convenience we define the following notations:
Φ(x,y)p˜(x+y)(1+ei(x+y)τ) g(x,y)(1+ei(x+y)τ) G(x,y). (5)
≡ ≡
The state produced by the down-conversionin the crystal reads
Ψ = i√I † vac + I † 2 vac + O(I3/2) (6)
| i A | i 2 A | i
(cid:0) (cid:1)
4
where I is proportional to the intensity I of the pump, and
p
† = dω dω G(ω ,ω )a†(ω )b†(ω ). (7)
a b a b a b
A Z
3.2 Detection: generalities
We have just given the state Ψ created by down-conversion. This state evolves through the setup (in
| i
our case, a linear optics one so that the number of photons is conserved) according to Ψ Ψˆ , then
| i → | i
two-photon coincidences are recorded. Here we introduce the general scheme for this detection. Let’s write
a and b the spatial modes on which one looks for coincidences; since no ambiguity is possibly, we write a
1 1 1
and b also the corresponding annihilation operators. We look at detector on mode a at time T ∆T,
1 1 A
±
where ∆T is the time resolution of the detectors; similarly for detection on mode b . The coincidence rate
1
reads
TA+∆T TB+∆T
R(T ,T ) = η η dt dt E(+)(t )E(+)(t ) Ψˆ 2. (8)
A B A B Z AZ B|| a1 A b1 B | i||
TA−∆T TB−∆T
In this formula, η are constant factors [18] that will be omitted in all that follows; the positive part of
A,B
the electric field on mode a is defined as
1
E(+)(t) = dνf (ν)e−iνta (ν). (9)
a1 Z A 1
withf (ν)isarealfunctiondescribingafilterinmodea ,thetransmissionofthefilterbeingF (ν)=f (ν)2.
A 1 A A
ThedefinitionofE(+)(t)isanalogous. Wechoosetheoriginoftimesinordertoremovethefreepropagation
b1
fromthecrystaltothedetectors. Therefore,the firsttime-binatthedetectionisgivenbyt =0,thesecond
j
time-bin by t =τ and so on.
j
Actually,formula(8)fordetectionisexactforproportionalcounters,inwhichtheprobabilityofdetectionis
theintensityofthe field. Forphotoncountingwithadetectorofquantumefficiencyη,theprobabilityofthe
detectorfiring,giventhatnphotonsimpingonit,isnotnη (proportionaltotheintensity)but(1 (1 η)n).
− −
Now, for the wavelengths that we consider, the quantum efficiency is η 0.1; moreover, the mean number
≈
of photons that imping on a detector is much smaller than 1 because of the losses in the fibers and in the
coupling; finally, in our formalism we restrict to the four-photon term, so that at most two photons can
imping on the detector. All in all, the approximation (1 (1 η)2) 2η holds and we can indeed use (8)
− − ≃
to compute the coincidence rate.
3.3 Important parameters
As we said in the introduction, we shall postpone the detailed calculations to the Appendix. All the results
of the Sections 4 and 5 can be formulated using the following parameters: writing dω =dω dω ,
a b
J = dω g(ω ,ω )2 , (10)
a b
Z | |
J = dωF (ω ) g(ω ,ω )2 (11)
A A a a b
Z | |
J = dωF (ω ) g(ω ,ω )2 (12)
B B b a b
Z | |
5
Figure 2: Schematic of the setup used to measure the parameter ρ.
J = dωF (ω )F (ω ) g(ω ,ω )2 , (13)
AB A a B b a b
Z | |
J = dωdω′F (ω )F (ω ) [g∗(ω ,ω )g∗(ω′,ω′)g(ω ,ω′)g(ω′,ω ) + c.c.] , (14)
4 Z A a B b a b a b a b a b
The first four numbers can be given an intuitive meaning. In fact, up to multiplicative factors: J is the
probability of producing two photons in one pump pulse, irrespective of whether they will pass the filter or
not; J and J are the probabilities of producing two photons in one pump pulse, and that the photon in
A B
mode a (resp. b) passes through the filter; J is the probability of producing two photons in one pump
AB
pulse and both photons pass the filter. The interpretationofJ is somehowmore involved: it is a coherence
4
term, due to the fact that the four photon state cannot be described as two independent pairs [9].
Obviously, J =J if no filter is applied on B. But J =J holds to a very good approximation also if
AB A AB A
∆ <∆ , where ∆ is the width of filter F , providedthat both filters are larger than the spectral width
A B X X
of the pump ∆ (as we supposed in Section 2, and as will be the case in the experiment). In fact, in this
p
case,detection ofa photonin filter A automaticallyensures thatits twin photonhas a frequency within the
range of filter B, which means F (ω )=1 for all ω compatible with the phase-matching condition.
B b b
4 A calibration setup
Before describing the measurement of two-photon interferences (next Section) we present an experimental
setup that allows to measure the probability ρ of creating a detectable pair in a simple way. This setup
(see Fig. 2) has been presented in detail in section IV of Ref. [6]. We give here a brief description. A
Fourier-transform-limited pulsed laser is used to create non-degenerate photon pairs at telecommunication
wavelengths (1310 and 1550 nm) by parametric down-conversion in a non linear-crystal. The two photons
areseparateddeterministicallyusingawavelength-divisionmultiplexer(WDM) andeachphotonisdetected
by single-photon counters (avalanche photodiodes). The signal from the two detectors are then sent to a
Time-to-Digital converter,which is used to determine the histogram of the differences in the time of arrival
of the twin photons.
6
Weapplyourformalismtothissetup. Forthedetection,sincethereisnoevolutionbutthefreepropagation,
we have simply a = a and b = b. For the preparation, at first sight it seems that our formalism should
1 1
be modified: we are dealing with a train of N pulses instead of only two pulses, so 1+eiωτ should be
replaced with N−1eiωkτ in formula (4). However, a closer look shows that we ca(cid:0)n do the(cid:1)calculation
k=0
P
without any change. In fact, in this particular setup there is no interference: then, R is simply the sum
C
of the coincidence rates obtained when the two photons arrive at the same time, while R is the sum of
L
the coincidence rates obtained when photon in mode a arrives a time τ later than the photon in mode b.
Since moreover R(kτ,kτ) = R(0,0) and R((k+1)τ,kτ) = R(τ,0) for all k, we obtain R = NR(0,0) and
C
R =(N 1)R(τ,0), so we can focus on only two successive pulses. By the way, R(0,0) is proportional to
C
−
the probability per pulse of creating one detectable pair (a pair that will pass the filters).
ThecalculationisgivenintheAppendix,andtheresultsareR(0,0) = IJ +O(I2)andR(τ,0) = I2J J ,
AB A B
that areindeed what one expects because of the meaning ofthe J’s (subsection3.3). Therefore,in the limit
of large N, the ratio ρ between the integrals of the side peak and the central peak is
J J
A B
ρ = I . (15)
J
AB
In most cases, ρ has a simple interpretation. In fact, whenever condition ∆ << ∆ < ∆ holds, we have
p A B
seen above that J = J and consequently ρ = IJ is the probability per pulse of creating a pair such
AB A B
that the photon that meets the largest filter will pass it. In particular, if there is no filter on mode b, ρ is
the probability per pulse of creating a pair, as noticed in the Appendix of [6]. That derivation shares with
the present one the hypothesis of small detector efficiency, but is otherwise rather different: in our previous
paper, we supposed that a 2N-photon state is actually N independent pairs; here, we limit ourselves to 2
and 4 photons, but derive the result without any assumption about the coherence of the 4-photon state.
Moreover, as argued in Section 2, since J J normally holds, at least in magnitude, then ρ IJ =
A B A
≃ ≃
R(0,0) is an estimate of the probability per pulse of creating a detectable pair.
5 The Franson interferometer
5.1 Description of the setup
Weturnnowtothemainsetup,whichistheinterferometerthatallowstheanalysisoftime-binentanglement
(see Fig.3). This is essentially the interferometer proposed by Franson to study energy-time entanglement
[19], completed with an unbalanced interferometer before the crystal (the pump interferometer). A laser
pulse is first split in two in this interferometer. At its exit, we have two laser pulses with a fixed phase
difference separated by a time τ corresponding to the path length difference between the long and the
short arm of the interferometer. In the non linear crystal, we therefore create a photon pair in a coherent
superposition of two time-bins.
After the crystal, the photons are separatedwith the WDM and eachsent to a fiber interferometer in order
to make a two photon interference experiment.
7
L
a
s
e
r
t
0
j
a nonlinear b
crystal
A B
+ +
A B
- -
&
Alice Bob
&
Figure3: Schematic ofthe setupusedtomeasuretwophotonquantuminterferencewithtime-binentangled
qubits. In addition to the two-photoncoincidence, a coincidence with the pump laser provides the origin of
time needed to define the three time-bins.
5.2 Evolution
The evolution of modes a and b in each arm of the interferometer is given by the following expressions:
a†(ω) aˆ†(ω) = S(ω,α)a†(ω)+C(ω,α)a†(ω) (16)
−→ 1 2
b†(ω) ˆb†(ω) = S(ω,β)b†(ω)+C(ω,β)b†(ω) (17)
−→ 1 2
with [20]
1 ei(ωτ+θ) 1+ei(ωτ+θ)
S(ω,θ) = − , C(ω,θ) = i . (18)
2 2
The evolved state Ψˆ is obtained by inserting the evolved operators aˆ† andˆb† into Ψ .
| i | i
In these formulae, we have already supposed that the analyzing interferometers are identical to the pump
interferometer. Thus, three time-bins are defined by the setup. The first time-bin, t=0, correspondsto the
timeofarrivalofphotonsproducedbythefirstpumppulseandnotdelayed. Thesecondorintermediatetime-
bin,t=τ,correspondstothetimeofarrival,eitherofphotonsproducedbythefirstpumppulseanddelayed,
or ofphotons producedby the secondpump pulse andnot delayed. The third time-bin, t=2τ, corresponds
to the time of arrival of photons produced by the second pump pulse and delayed. Interferences will only
be seen when both photons arrive at t = τ, because only in this case two indistinguishable alternatives are
available.
5.3 Two-photon interferences
We study the detection for modes a and b ; all the other cases can be treated in the same way. The
1 1
coincidence rate R(T ,T ) is the sum of two terms corresponding respectively to the two-photon and the
A B
8
four-photon terms:
R (T ,T ) = I dt dt E(+)(t )E(+)(t ) ˆ† vac 2, (19)
2 A B Z A B|| a1 A b1 B A | i||
R (T ,T ) = I2 dt dt E(+)(t )E(+)(t ) ˆ† 2 vac 2. (20)
4 A B 4 Z A B|| a1 A b1 B A | i||
(cid:0) (cid:1)
Indeed, the two-andthe four-photonstates do notinterfere(in principle, one couldinserta non-destructive
measurement of the number of photons just after the crystal, and this would not modify the rest of the
experiment).
The calculation is presented in the Appendix. As said above, interferences will appear only in the interme-
diate time-bin T =T =τ, in which case one finds [21]:
A B
R (τ,τ) = IJ 1+cos(α+β) , (21)
2 AB
(cid:0) (cid:1)
R (τ,τ) = I2 2J J + J ) 1+cos(α+β) + 2J J . (22)
4 AB 4 A B
h(cid:0) (cid:0) (cid:1) i
The result for R (τ,τ) is the expected one: one pair is produced, it passes the filters, and since it is in a
2
superposition of being in both pulses it gives rise to full-visibility interferences. In the formula for R (τ,τ),
4
twocontributionsarealsoexpectedfromtheintuitiveviewofthefour-photonstateastwoindependentpairs:
(i) the term containing J J means that two pairs are created, the photons of the same pair are detected
AB
and therefore one has full visibility; (ii) the term containing J J means that two pairs are created, the
A B
photons of the different pair are detected and therefore they don’t show any interference. The remarkable
feature is the position of the correction due to the coherence in the four-photon term, J : it contributes to
4
a full-visibility interference as well. This couldn’t have been guessed without the full calculation.
Summing (21) and (22), the total two-photon coincidence rate in the intermediate time-bin reads
R(τ,τ) = R (τ,τ)+R (τ,τ) = R¯ 1 + V cos(α+β) (23)
2 4
(cid:2) (cid:3)
where the average count rate R¯ is given by [21]
R¯ =IJ + O(I2) (24)
AB
and the visibility V is given by V = 1+I(J+J4/JAB) . Now, the terms O(I2) in the visibility are
1+I(J+(J4+JAJB)/JAB)
meaningless,becausethe six-photontermthatweneglectedcompletelycontributestothe sameorder;sowe
have to keep only the first-order development of V in I, that leads to the remarkable relation
2J J
A B
V 1 I = 1 2ρ (25)
≃ − J −
AB
where ρ is exactly the same as defined in (15). As announced in Section 2, J drops out of the visibility: to
4
the leading order in I, the loss of visibility is independent of the coherence of the four-photon state.
Since ρ is basically the probability per pulse (so that 2ρ is the probability per qubit [14]) of creating a
detectablepair,itdefinesthe detectionrateuptomultiplicativefactors. Relation(25)thereforesaysthat,if
we fix a detectionrate,we shallfind a givenvisibility, no matter whether the ratewasobtainedby pumping
weakly and putting no filters, or by pumping strongly and putting narrow filters. This is a positive feature:
filters, while being useful to improve the coherence whenever this is required, do not degrade the visibility.
9
As described in section 4, ρ can be measured independently, the relation (25) can be experimentally tested.
This is the object of the next Section.
6 Experimental verification
In this section, we present an experimental verification of Eq. (25). Two-photon interference fringes are
recorded for different value of ρ, corresponding to different values of pump power, with the Franson setup
described in the previous Section. Let us remind the reader that the down-converted photons are at the
two telecom wavelengths, 1310 nm and 1550 nm. The measurement are reported for two different filters
configurations; in both cases, the larger filter is on the photons at 1550nm, so this is ”mode b”.
In the first configuration, only the photon at 1310 nm is filtered with 40 nm FWHM. These data are taken
from[6]. Inthesecondconfiguration,bothphotonarefiltered. Thephotonat1310nmisfilteredwith10nm
FWHM,while the photonat1550nmis filteredwith18nmFWHM.The coefficientρis measuredusingthe
side peaks method explained in Section 4. The visibility for the two experimental configurations is plotted
as a function of 2ρ in Fig. 4. The error bars on the experimental points represent the accuracy of the fit of
the recorded interference patterns with a sine law [22]. The two solid lines are straight lines with slope 1,
−
according to Eq. (25); the small shift between the two curves is due to the fact that the maximal visibility
wasnotthesameforbothexperimentsandwasleftfreeasafittingparameter. Weobserveagoodagreement
betweentheoryandexperiment. Theseresultsconfirmthatthelossofvisibility dueto four-photoneventsis
directlyrelatedtoρ,regardlessofthefilteringthatisappliedonthe photonsandregardlessofthecoherence
of the four-photon component [9]. This is therefore a general result very useful to estimate the effect of
multi-pair creation in an experiment in a very simple way.
7 Conclusion
Insummary,we havefound a quantitativepredictionfor the loss oftwo-photoninterferencevisibility due to
thepresenceofafour-photoncomponentinthedown-convertedfield. Thelossofvisibility(25)isdetermined
by the parameter ρ (15), that is close to the probability of creating a detectable pair. This parameter can
bemeasuredindependently,thusallowingadirectexperimentalverificationofourprediction. Whilethe full
calculationwasworkedoutfortime-binentanglement,wehavepresentedinSection2asimplifiedderivation
that gives the same result and applies to any form of entanglement generated by down-conversion.
We acknowledge fruitful discussions with Antonio Ac´ın, Christoph Simon and Wolfgang Tittel.
Note added in proof. Since this work was finished, we have learnt of two independent papers [23, 24] that
discussed the loss of visibility of two-photon interferences due to the presence of higher-photon-number
components. Bothcalculationsconcernentanglementinpolarizationandhavebeendone inthe single-mode
formalism: this allows to take into account the contribution of all more-photon terms and not only of the
four-photon one. The results are compatible with ours in the regime where they can be compared (small
pump power). Consider for instance Ref. [24]: from their eq. (2), we see that the probability per qubit of
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